Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,5,8,7)(10,13)(11,15)(18,22)(20,24)(25,26,30,29)(27,28,32,31);(1,3,6,8)(2,7,4,5)(11,15)(14,16)(18,22)(19,23)(25,27,30,32)(26,31,29,28);(1,2,6,4)(3,5,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,20,23,24)(25,27,30,32)(26,31,29,28);(1,4,6,2)(3,7,8,5)(9,11)(10,14)(12,15)(13,16)(17,19,21,23)(18,20,22,24)(25,28)(26,27)(29,32)(30,31);(1,5,6,7)(2,3,4,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,22)(21,24)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3822848);